Imaginary Numbers

What is the square root of a negative number? Did you know that no real number multiplied by itself will ever produce a negative number? Finding the square root of 4 is simple enough: either 2 or -2 multiplied by itself gives 4. However, there is no simple answer for the square root of -4.

So, what do you do when a discriminant is negative and you have to take its square root? This is where imaginary numbers come into play. Essentially, mathematicians have decided that the square root of -1 should be represented by the letter i. So, i = sqrt(-1), or you can write it this way: -1 1/2 or you can simply say: i 2 = -1.

What you should know about the number i:

1) i is not a variable.
2) i is not found on the real number line.
3) i is not a real number.

Sample A:

Simplify (4i) 2

Steps:

1) Multiply 4i times 4i. This will produce 16(i 2 ).

2) Multiply 16 times -1 because i2 equals -1.

The answer is: -16.

Sample B:

Simplify sqrt(-80).

Steps:

1) Multiply two radicands keeping in mind that one of them has to be a perfect square. How about sqrt(16) times sqrt(5)? Yes, this will produce sqrt(80). Also, don't forget to multiply sqrt(-1) times sqrt(16) times sqrt(5).

2) Simplify square roots where needed. For example, sqrt(16) becomes simply 4 and sqrt-1 simply becomes the number i.

3) Put it all together this way: 4i(sqrt(5)) or 4i times the square root of 5.

NOTE: You cannot reduce sqrt5 anymore because it is already in lowest terms.

Here's another lesson on imaginary numbers if you would like another view point.

By Mr. Feliz
(c) 2005